Question - Converting a Repeating Decimal to a Fraction
mathfraction simplificationconverting decimal fractionrepeating decimal to fractiondecimal fraction conversion
Solution:
The image displays a handwritten mathematical question which asks to express the number 0.282323... in the form $$\frac{a}{b}$$, where $$a$$ and $$b$$ are integers.The given number is a repeating decimal: 0.282323... with the pattern "23" repeating indefinitely.To convert this repeating decimal to a fraction, follow these steps:1. Let $$x = 0.282323...$$2. Notice that the repeating block "23" starts after the hundredths place.3. To shift the repeating block to start immediately after the decimal, multiply $$x$$ by 1000: $$1000x = 282.323...$$4. Now subtract $$x$$ from $$1000x$$ to get a number where the decimal part cancels out.5. $$1000x - x = 282.323... - 0.282323...$$6. $$999x = 282$$7. Divide by 999 to solve for $$x$$: $$x = \frac{282}{999}$$8. Simplify the fraction: $$\frac{282}{999} = \frac{94}{333}$$ (dividing numerator and denominator by 3)The answer is $$x = \frac{94}{333}$$.
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